Consider the singular porous medium equation $$u_t - \Delta (|u|^{m-1}u) = 0$$ $$u(0)=u_0$$ given $u_0$ bounded, where $m \in (0,1)$.

When posed on $\mathbb{R}^n$, it is well known that mass is conserved when $m \in (m_c, 1)$ and mass is lost when $m \in (0,m_c)$, where $m_c$ is a number that depends on the dimension $n$.

When we pose this equation on a $n-$dimensional compact closed manifold, does this still hold true? Do we still get a range for which mass is lost, i.e., $$\int_M u(t,x)dx \neq \int_M u_0?$$

I've tried to find papers but there is not much on manifolds.